 a a + 2kN fa .


Given this symmetry, \({\mathcal {N}}\)  mirror and degenerate worlds linked by the axion field would coexist with the same coupling strengths as in the SM, with the exception of the
effective \(\theta _k\) -parameters: for each copy \(k\)  the effective \(\theta \) -value is shifted by \(2\pi /{\mathcal {N}}\)  with respect to that in the
neighboring \(k-1\)  sector. Thus the total potential for the axion is given by the sum of all the shifted contributions,

VN(a)=-m2 f2k=0N-1 1-4 mu md(mu+md)2 2(a2+kN)  ,


where \(\theta _a \equiv a / f_a\)  is the angular axion field.
Strikingly, the resulting axion is exponentially lighter than the canonical QCD axion in Eq. (), because the non-perturbative contributions to its potential from the \({\mathcal {N}}\)  degenerate QCD groups conspire by symmetry to suppress each other [1]}, [2]}.
Indeed, it has been shown that this kind of \(Z_{\mathcal {N}}\)  symmetric potentials have interesting mathematical propertiesSee Ref. [3]} for a generalization of the mechanism for non-abelian discrete symmetries. in the large \({\mathcal {N}}\)  limit and for this case the total axion potential is given in all generality by a compact analytical formula [2]},


VN(a)
- ma2 fa2N2  (Na) ,       ma2   f2a m2 f2  1-z1+z  N3/2  zN ,

where the exponential suppression of the \(Z_{\mathcal {N}}\)  axion mass squared (\(\propto 2^{-{\mathcal {N}}}\) ) in comparison to the canonical case \(\left(m_a^{\rm QCD}\right)^2\)  in Eq. () is controlled by the ratio of light quark masses \(z\equiv m_u/m_d\simeq 1/2\) .
